A quantum-inspired classical algorithm for recommendation systems

Abstract

We give a classical analogue to Kerenidis and Prakash's quantum recommendation system, previously believed to be one of the strongest candidates for provably exponential speedups in quantum machine learning. Our main result is an algorithm that, given an m × n matrix in a data structure supporting certain ℓ 2 -norm sampling operations, outputs an ℓ 2 -norm sample from a rank-k approximation of that matrix in time O(poly(k) log(mn)), only polynomially slower than the quantum algorithm. As a consequence, Kerenidis and Prakash's algorithm does not in fact give an exponential speedup over classical algorithms. Further, under strong input assumptions, the classical recommendation system resulting from our algorithm produces recommendations exponentially faster than previous classical systems, which run in time linear in m and n.The main insight of this work is the use of simple routines to manipulate ℓ 2 -norm sampling distributions, which play the role of quantum superpositions in the classical setting. This correspondence indicates a potentially fruitful framework for formally comparing quantum machine learning algorithms to classical machine learning algorithms.

Challenges and Opportunities of Near-Term Quantum Computing Systems

“…Not all quantum circuits can lead to a quantum advantage. If the quantum algorithm can be efficiently simulated on classical hardware [110]- [116], it can not provide a computational advantage. The advantage of quantum computers is based on the complexity of the algorithm and not on the quantum computers ability to perform fast operations.…”

A Quantum-inspired Classical Algorithm for Separable Non-negative Matrix Factorization

“…QDCA implements matrix-vector product (i.e., random projection) via quantum principal component analysis and then a quantum state encoding the projected data points could be prepared efficiently. Moreover, there are also several papers utilizing dequantizing techniques to solve some low-rank matrix operations, such as recommendation systems [Tan18] and matrix inversion [GLT18,CLW18]. Dequantizing techniques in those algorithms involve two technologies, the Monte-Carlo singular value decomposition and rejection sampling, which could efficiently simulate some special operations on low-rank matrices.…”

“…. Given a description ofV, we can sample fromV (i) in time O(poly(k, 1 )) for i ∈ [k] [Tan18] and query its entry in time O(poly(k, 1 )).…”

“…Lemma 2 (Properties of α-orthonormal Vectors, [Tan18]). Given a set of k α-approximately orthonormal vectorsV ∈ R n×k , then there exists a set of k orthonormal vectors V ∈ R n×k spanning the columns ofV such that…”

“…Lemma 6 ([Tan18]). Given a set of α-approximately orthonormal vectorsV ∈ R n×k in vector sample model, and an input vector w ∈ R k , there exists an algorithm outputting a sample from a distribution α 1−α -close to DV w with probability 1 − γ using O(k 2 log 1 γ (1 + O(α))) queries and samples.Lemma 7.…”